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1、第六讲 极大似然估计The Likelihood Function and Identification of the Parameters (极大似然函数及参数识别) 1、似然函数的表示在具有n个观察值的随机样本中，每个观察值的密度函数为。由于n个随机观察值是独立的，其联合密度函数为函数被称为似然函数，通常记为，或者。与Greene书中定义的区别The probability density function, or pdf for a random variable y, conditioned on a set of parameters, , is denoted . This fun
2、ction identifies the data generating process that underlies an observed sample of data and, at the same time, provides a mathematical description of the data that the process will produce. The joint density of n independent and identically distributed (iid) observations from this process is the prod
3、uct of the individual densities; (17-1)This joint density is the likelihood function, defined as a function of the unknown parameter vector, , where is used to indicate the collection of sample data. Note that we write the joint density as a function of the data conditioned on the parameters whereas
4、 when we form the likelihood function, we write this function in reverse, as a function of the parameters, conditioned on the data. Though the two functions are the same, it is to be emphasized that the likelihood function is written in this fashion to highlight our interest in the parameters and th
5、e information about them that is contained in the observed data. However, it is understood that the likelihood function is not meant to represent a probability density for the parameters as it is in Section 16.2.2. In this classical estimation framework, the parameters are assumed to be fixed consta
6、nts which we hope to learn about from the data.It is usually simpler to work with the log of the likelihood function:. (17-2)Again, to emphasize our interest in the parameters, given the observed data, we denote this function . The likelihood function and its logarithm, evaluated at , are sometimes
7、denoted simply and , respectively or, where no ambiguity can arise, just or .It will usually be necessary to generalize the concept of the likelihood function to allow the density to depend on other conditioning variables. To jump immediately to one of our central applications, suppose the disturban
8、ce in the classical linear regression model is normally distributed. Then, conditioned on its specific is normally distributed with mean and variance . That means that the observed random variables are not iid; they have different means. Nonetheless, the observations are independent, and as we will
9、examine in closer detail, (17-3)where is the matrix of data with row equal to .2、识别问题The rest of this chapter will be concerned with obtaining estimates of the parameters, and in testing hypotheses about them and about the data generating process. Before we begin that study, we consider the question
10、 of whether estimation of the parameters is possible at allthe question of identification. Identification is an issue related to the formulation of the model. The issue of identification must be resolved before estimation can even be considered. The question posed is essentially this: Suppose we had
11、 an infinitely large samplethat is, for current purposes, all the information there is to be had about the parameters. Could we uniquely determine the values of from such a sample? As will be clear shortly, the answer is sometimes no. 注意：希望大家能够熟练地写出不同分布的密度函数，以及对应的似然函数。这是微观计量经济学的基本功。特别是正态分布、Logistic分
12、布。更一般地讲，指数类分布的密度函数。 17.3 Efficient estimation: the Principle of Maximum Likelihood The principle of maximum likelihood provides a means of choosing an asymptotically efficient estimator for a parameter or a set of parameters.The logic of the technique is easily illustrated in the setting of a discre
13、te distribution. Consider a random sample of the following 10 observations from a Poisson distribution: 5, 0, 1, 1, 0, 3, 2, 3, 4, and 1. The density for each observation is Since the observations are independent, their joint density, which is the likelihood for this sample, is.The last result gives
14、 the probability of observing this particular sample, assuming that a Poisson distribution with as yet unknown parameter generated the data. What value of would make this sample most probable? Figure 17.1 plots this function for various values of. It has a single mode at , which would be the maximum
15、 likelihood estimate, or MLE, of .Consider maximizing with respect to . Since the log function is monotonically increasing and easier to work with, we usually maximize instead; in sampling from a Poisson population,For the assumed sample of observations,andThe solution is the same as before. Figure 17.1 also plots the log of to illustrate the result
